Editing Transfer function (section)

==Signal processing==
If <math> x(t) </math> is the input to a general [[LTI system theory|linear time-invariant system]], and <math> y(t) </math> is the output, and the [[bilateral Laplace transform]] of <math> x(t) </math> and <math> y(t) </math> is

: <math>\begin{align}
  X(s) &= \mathcal{L}\left \{ x(t) \right \} \ \stackrel{\mathrm{def}}{=}\  \int_{-\infty}^{\infty} x(t) e^{-st}\, dt, \\
  Y(s) &= \mathcal{L}\left \{ y(t) \right \} \ \stackrel{\mathrm{def}}{=}\  \int_{-\infty}^{\infty} y(t) e^{-st}\, dt.
\end{align}</math>

The output is related to the input by the transfer function <math> H(s) </math> as

: <math> Y(s) = H(s) X(s) </math>

and the transfer function itself is 

: <math> H(s) = \frac{Y(s)} {X(s)}. </math>

If a [[complex number|complex]] [[harmonic]] [[signal (information theory)|signal]] with a [[sinusoidal]] component with [[amplitude]] <math>|X|</math>, [[angular frequency]] <math>\omega</math> and [[Phase (waves)|phase]] <math>\arg(X)</math>, where arg is the [[Argument (complex analysis)|argument]]

:<math> x(t) = Xe^{j\omega t} = |X|e^{j(\omega t + \arg(X))} </math>

:where <math> X = |X|e^{j\arg(X)} </math>

is input to a [[linear]] time-invariant system, the corresponding component in the output is:

:<math>\begin{align}
  y(t) &= Ye^{j\omega t} = |Y|e^{j(\omega t + \arg(Y))}, \\
     Y &= |Y|e^{j\arg(Y)}.
\end{align}</math>

In a linear time-invariant system, the input frequency <math> \omega </math> has not changed; only the amplitude and phase angle of the sinusoid have been changed by the system. The [[frequency response]] <math> H(j \omega) </math> describes this change for every frequency <math> \omega </math> in terms of gain

:<math>G(\omega) = \frac{|Y|}{|X|} = |H(j \omega)| </math>

and phase shift

:<math>\phi(\omega) =  \arg(Y) -  \arg(X) = \arg( H(j \omega)).</math>

The [[phase delay]] (the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is

:<math>\tau_{\phi}(\omega) = -\frac{\phi(\omega)}{\omega}.</math>

The [[group delay]] (the frequency-dependent amount of delay introduced to the envelope of the sinusoid by the transfer function) is found by computing the derivative of the phase shift with respect to angular frequency <math> \omega </math>,

:<math>\tau_{g}(\omega) = -\frac{d\phi(\omega)}{d\omega}.</math>

The transfer function can also be shown using the [[Fourier transform]], a special case of [[bilateral Laplace transform]] where <math> s = j \omega </math>.

=== {{anchor|Common transfer function families}}Common transfer-function families ===

Although any LTI system can be described by some transfer function, "families" of special transfer functions are commonly used:
* [[Butterworth filter]]&nbsp;– maximally flat in passband and stopband for the given order
* [[Chebyshev filter|Chebyshev filter (Type I)]]&nbsp;– maximally flat in stopband, sharper cutoff than a Butterworth filter of the same order
* Chebyshev filter (Type II)&nbsp;– maximally flat in passband, sharper cutoff than a Butterworth filter of the same order
* [[Bessel filter]]&nbsp;– maximally constant [[group delay]] for a given order
* [[Elliptic filter]]&nbsp;– sharpest cutoff (narrowest transition between passband and stopband) for the given order
* [[Optimum "L" filter]]
* [[Gaussian filter]]&nbsp;– minimum group delay; gives no overshoot to a step function
* [[Raised-cosine filter]]